Question: Simplify; express your answer in exponential form. Assume $k\neq 0, z\neq 0$. $\dfrac{{(k^{5}z^{-2})^{-5}}}{{(kz^{3})^{-3}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(k^{5}z^{-2})^{-5} = (k^{5})^{-5}(z^{-2})^{-5}}$ On the left, we have ${k^{5}}$ to the exponent ${-5}$ . Now ${5 \times -5 = -25}$ , so ${(k^{5})^{-5} = k^{-25}}$ Apply the ideas above to simplify the equation. $\dfrac{{(k^{5}z^{-2})^{-5}}}{{(kz^{3})^{-3}}} = \dfrac{{k^{-25}z^{10}}}{{k^{-3}z^{-9}}}$ Break up the equation by variable and simplify. $\dfrac{{k^{-25}z^{10}}}{{k^{-3}z^{-9}}} = \dfrac{{k^{-25}}}{{k^{-3}}} \cdot \dfrac{{z^{10}}}{{z^{-9}}} = k^{{-25} - {(-3)}} \cdot z^{{10} - {(-9)}} = k^{-22}z^{19}$